# Publications 2018

** Bhatt, A.; Fehr, J. & Hassdonk, B.:**
Model Order Reduction of an Elastic Body under Large Rigid Motion,
*Proceedings of ENUMATH 2017, Voss, Norway, *
**2018**.

@inproceedings
{Bhatt2018,

author = {Bhatt, A. and Fehr, J. and Hassdonk, B.}
,

title = {Model Order Reduction of an Elastic Body under Large Rigid Motion}
,

booktitle = {Proceedings of ENUMATH 2017, Voss, Norway}
,

year = {2018}
}

** Brünnette, T.; Santin, G. & Haasdonk, B.:**
Greedy kernel methods for accelerating implicit integrators for parametric ODEs,
*University of Stuttgart, *
**2018***, Proceedings of ENUMATH 2017*.

@inproceedings
{Bruennette_Enumath2017,

author = {Brünnette, Tim and Santin, Gabriele and Haasdonk, Bernard}
,

title = {Greedy kernel methods for accelerating implicit integrators for parametric ODEs}
,

year = {2018}
,

volume = {Proceedings of ENUMATH 2017}
,

url = {http://www.simtech.uni-stuttgart.de/publikationen/prints.php?ID=1767}
}

**Abstract: ** We present a novel acceleration method for the solution of parametric ODEs by single-step implicit solvers by means of greedy kernel-based surrogate models. In an offline phase, a set of trajectories is precomputed with a high-accuracy ODE solver for a selected set of parameter samples, and used to train a kernel model which predicts the next point in the trajectory as a function of the last one. This model is cheap to evaluate, and it is used in an online phase for new parameter samples to provide a good initialization point for the nonlinear solver of the implicit integrator. The accuracy of the surrogate reflects into a reduction of the number of iterations until convergence of the solver, thus providing an overall speedup of the full simulation. Interestingly, in addition to providing an acceleration, the accuracy of the solution is maintained, since the ODE solver is still used to guarantee the required precision. Although the method can be applied to a large variety of solvers and different ODEs, we will present in details its use with the Implicit Euler method for the solution of the Burgers equation, which results to be a meaningful test case to demonstrate the method's features.

** De Marchi, S.; Iske, A. & Santin, G.:**
Image reconstruction from scattered Radon data by weighted positive definite kernel functions,
*Calcolo, *
**2018***, 55*, 2.

@article
{DeMarchi2018,

author = {De Marchi, S. and Iske, A. and Santin, G.}
,

title = {Image reconstruction from scattered Radon data by weighted positive definite kernel functions}
,

journal = {Calcolo}
,

year = {2018}
,

volume = {55}
,

number = {1}
,

pages = {2}
,

url = {https://doi.org/10.1007/s10092-018-0247-6}
,

doi = {10.1007/s10092-018-0247-6}
}

**Abstract: ** We propose a novel kernel-based method for image reconstruction from scattered Radon data. To this end, we employ generalized Hermite--Birkhoff interpolation by positive definite kernel functions. For radial kernels, however, a straightforward application of the generalized Hermite--Birkhoff interpolation method fails to work, as we prove in this paper. To obtain a well-posed reconstruction scheme for scattered Radon data, we introduce a new class of weighted positive definite kernels, which are symmetric but not radially symmetric. By our construction, the resulting weighted kernels are combinations of radial positive definite kernels and positive weight functions. This yields very flexible image reconstruction methods, which work for arbitrary distributions of Radon lines. We develop suitable representations for the weighted basis functions and the symmetric positive definite kernel matrices that are resulting from the proposed reconstruction scheme. For the relevant special case, where Gaussian radial kernels are combined with Gaussian weights, explicit formulae for the weighted Gaussian basis functions and the kernel matrices are given. Supporting numerical examples are finally presented.

** Fritzen, F.; Haasdonk, B.; Ryckelynck, D. & Schöps, S.:**
An algorithmic comparison of the Hyper-Reduction and the Discrete Empirical Interpolation Method for a nonlinear thermal problem,
*Math. Comput. Appl. 2018, *
*University of Stuttgart, *
**2018***, 23*.

@article
{FHRS18,

author = {Felix Fritzen and Bernard Haasdonk and David Ryckelynck and Sebastian Schöps}
,

title = {An algorithmic comparison of the Hyper-Reduction and the Discrete Empirical Interpolation Method for a nonlinear thermal problem}
,

journal = {Math. Comput. Appl. 2018}
,

year = {2018}
,

volume = {23}
,

number = {1}
,

url = {http://www.mdpi.com/2297-8747/23/1/8/pdf}
,

doi = {10.3390/mca23010008}
}

** Haasdonk, B. & Santin, G.:**
*Keiper, Winfried and Milde, Anja and Volkwein, Stefan **(Eds.)*,
Greedy Kernel Approximation for Sparse Surrogate Modeling,
*Reduced-Order Modeling (ROM) for Simulation and Optimization: Powerful Algorithms as Key Enablers for Scientific Computing, *
*Springer International Publishing, *
**2018**, 21-45.

@inbook
{HS2017a,

author = {Haasdonk, Bernard and Santin, Gabriele}
,

editor = {Keiper, Winfried and Milde, Anja and Volkwein, Stefan}
,

title = {Greedy Kernel Approximation for Sparse Surrogate Modeling}
,

booktitle = {Reduced-Order Modeling (ROM) for Simulation and Optimization: Powerful Algorithms as Key Enablers for Scientific Computing}
,

publisher = {Springer International Publishing}
,

year = {2018}
,

pages = {21--45}
,

url = {https://doi.org/10.1007/978-3-319-75319-5_2}
,

doi = {10.1007/978-3-319-75319-5_2}
}

**Abstract: ** Modern simulation scenarios frequently require multi-query or real-time responses of simulation models for statistical analysis, optimization, or process control. However, the underlying simulation models may be very time-consuming rendering the simulation task difficult or infeasible. This motivates the need for rapidly computable surrogate models. We address the case of surrogate modeling of functions from vectorial input to vectorial output spaces. These appear, for instance, in simulation of coupled models or in the case of approximating general input--output maps. We review some recent methods and theoretical results in the field of greedy kernel approximation schemes. In particular, we recall the vectorial kernel orthogonal greedy algorithm (VKOGA) for approximating vector-valued functions. We collect some recent convergence statements that provide sound foundation for these algorithms, in particular quasi-optimal convergence rates in case of kernels inducing Sobolev spaces. We provide some initial experiments that can be obtained with non-symmetric greedy kernel approximation schemes. The results indicate better stability and overall more accurate models in situations where the input data locations are not equally distributed.

** Köppl, T.; Santin, G.; Haasdonk, B. & Helmig, R.:**
Numerical modelling of a peripheral arterial stenosis using dimensionally reduced models and kernel methods,
*International Journal for Numerical Methods in Biomedical Engineering, *
**2018***, 0*, e3095.

@article
{KSHH2017,

author = {Köppl, Tobias and Santin, Gabriele and Haasdonk, Bernard and Helmig, Rainer}
,

title = {Numerical modelling of a peripheral arterial stenosis using dimensionally reduced models and kernel methods}
,

journal = {International Journal for Numerical Methods in Biomedical Engineering}
,

year = {2018}
,

volume = {0}
,

number = {ja}
,

pages = {e3095}
,

note = {e3095 cnm.3095}
,

url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/cnm.3095}
,

doi = {10.1002/cnm.3095}
}

**Abstract: ** Summary In this work, we consider two kinds of model reduction techniquesto simulate blood flow through the largest systemic arteries, where a stenosis is located in a peripheral artery i.e. in an artery that is located far away from the heart. For our simulations we place the stenosis in one of the tibial arteries belonging to the right lower leg (right post tibial artery). The model reduction techniques that are used are on the one hand dimensionally reduced models (1‐Dand 0‐D models, the so‐called mixed‐dimension model) and on the other hand surrogate models produced by kernel methods. Both methods are combined in such a way that the mixed‐dimension models yield training data for the surrogate model, where the surrogate model is parametrisedby the degree of narrowing of the peripheral stenosis. By means of a well‐trained surrogate model, we show that simulation data can be reproduced with a satisfactory accuracy and that parameter optimisation or state estimation problems can be solved in a very efficient way. Furthermore it is demonstrated that a surrogate model enables us to present after a very short simulation time the impact of a varying degree of stenosis on blood flow, obtaining a speedup of several orders over the full model.